Most
students understand that the probability of an event occurring can be
influenced by another event that has already occurred.However, many students cannot understand that the probability of an
event occurring can actually be dependent on an event that occurred later.Having information about the outcome of an event can be used to
revise probabilities of the occurrence of a previous event.This lesson plan will help teachers to correct this common student
misconception.For students with more probability experience, this lesson also
introduces them to Bayesí Theorem.Bayesí Theorem provides a formula that to find one conditional
probability if other conditional probabilities are known.More specifically, it can be used to find P(A|B) if P(B|A) is
known.Bayesí
Theorem is usually expressed as:

III. Rationale: To correct a common
misconception about probability (known as the Time-Axis
Fallacy) and to
introduce Bayes' Theorem.

IV. Learning Objectives:

1. Students will learn that the knowledge of
an eventís outcome can affect the probability of the unknown outcome of
an event that has already occurred.

2. Students will learn how to apply
Bayes' Theorem to
solve probability problems.

V. Materials & Technology Needed:

1. Bag and four jellybeans for modeling (optional).

2. Calculators for mathematical computations.

VI. Procedure:

PART
1: INTRODUCTION

1.
Present the following situations to your students and ask them the
questions.For all situations, suppose that there are four jellybeans left in
a bag.Two
are grape and two are licorice

2.
Situation 1:If I reach in to the bag without looking and choose a jellybean at
random, what is the probability that it is a grape one?What is the probability that I choose a licorice jellybean?

3.
Situation 1 (continued):Suppose that I choose a grape jellybean.I reach into the bag again to choose another jellybean at random.What is the probability that I pull out the other grape one?What is the probability that I take a licorice one?Why do the probabilities change from what they were before?

PART
2: TREE DIAGRAM

4.
Represent this situation as a tree diagram.(For students who are unfamiliar with tree diagrams, some
explanation and terminology might be necessary.For the rest of this lesson, I will refer to the nodes and the
branches of tree diagrams.For every event that has more than one possible outcome, the tree
diagram will have a node.Any possible outcome at a node is represented by a branch.)

5.
What does the first node of the diagram represent?How many branches does it have?How do we label them? What
probabilities do we assign to each branch?

6.
Follow the branch that assumes that the first jellybean I take is grape.What does the node at the end of this branch represent? How
many branches does it have? How
do we label them? What
probabilities do we assign to each branch?

7.
Repeat these steps for the other branch of the first node (the one that
assumes that the first jellybean I take is licorice).

8.
The tree diagram now has four possible final outcomes.Label them.How can you find the probability associated with each of these
branches?

9.
Suppose I start back at the beginning, with two grape jellybeans and two
licorice jellybeans in the bag.Situation 2: This time I choose one jellybean at random and put it
in my pocket without looking at it.I choose another jellybean at random and it is grape-flavored.What is the probability that the jellybean in my pocket is also
grape?What
is the probability that it is a licorice jellybean?

10.
Why isnít it equally likely that the jellybean in my pocket is either
flavor when there were two of each flavor in the bag when I took it?

11.
Letís use the formula P(AB) = P(A|B) * P(B) to find these probabilities
(the ones from Procedure 9).What probabilities do we want to know?What else do we need to use the formula? Which
of these do we already know? How
can we find P(G2) from the tree diagram? Now,
use the formula.

12.
Letís use the tree diagram to help us find these probabilities.Are there any branches that are not necessary in this situation
(Situation 2)?Why? What
branches are still possible?Is there a special relationship between these two events? What
is the relationship between the probabilities of these events? What
probabilities can we assign to two complementary events where one event is
twice as likely to occur as the other event?

PART 3: BAYES' THEOREM

13. Thereís a mathematical formula that can help us
answer this kind of question called Bayes' Theorem.This formula is most useful when we need to find P(A|B) but we can
only easily find P(B|A) and P(A).

Formula from Bayes' Theorem:

P(AB)

[P(B|A)
* P(A)]

P(A|B)

=

-----------

=

--------------------------------------------

P(B)

[P(B|A) * P(A) +
P(B|~A) * P(~A)]

14. Referring back to the second jellybean situation,
what probabilities are we looking for? What
probabilities do we know from the Tree Diagram?Use Bayes' Theorem to find the two probabilities weíre
looking for.Are the answers
the same as we found from the Tree Diagram?

15. A common application of
Bayes' Theorem is medical
testing for diseases.Doctors
perform tests to help determine if their patients have certain diseases.The tests indicate that the person either has the disease (a
positive result) or does not have it (a negative result).Based on studies, doctors know the probability of a random person
from a population having a particular disease.They also know the probabilities of a person with the disease
getting a positive test result and of a person without the disease getting
a negative test result.

16. In the fictional town of Atlantis, doctors have
developed a test for the Atlantis Death Flu.They know that the probability that a random person in Atlantis has
the Atlantis Death Flu is .01.They
also know a person who has the Atlantis Death Flu has a .95 probability of
testing positive for the disease while a person who does not have it has a
.94 probability of getting a negative test result.Letís say that a patient comes to a doctor because he wants to
get tested for Atlantis Death Flu.What
is the probability that he actually has Atlantis Death Flu is he tests
positively for it?

17. Letís represent this situation with a tree
diagram.What should appear
first in the tree diagram: whether or not a person has the disease or
whether or not a person tests positive for the disease?Why?How do we label
the two branches of the first node?What
probabilities go with each branch?How
do we know the probability that somebody in Atlantis does not have the
Atlantis Death Flu?

18. Follow the branch that assumes that a person has
the disease.How do we label
the two branches of this node?What
probabilities go with each branch?How
do we know the probability that somebody with the Atlantis Death Flu tests
negative?

19. Repeat these steps with the node at the end of
the branch that assumes that a person does not have the Atlantis Death
Flu.

20. What probability are we looking for?What probabilities do we know from the Tree Diagram?Use Bayes' Theorem to answer the question.

VII. Assessment:

1. Assume I have a normal deck of cards.I draw a random card from it and place it face down on the table
without looking at it.Then I
draw a second card and I see that it is a red card.

(a). Explain how I now have more information about
the color of the first card because I know what color the second card is.

(b). What is the probability that the first card is
red?What is the probability
that the first card is black?

(c). Draw a Tree Diagram to represent this situation.Label your diagram, and if you use any abbreviations, explain what
they stand for.

(d). Use Bayes' Theorem to check the probabilities
you derived in part (a).

2. Use Bayes' Theorem and the probabilities from the
Atlantis Death Flu situation to find P(~A|+) and P(A|-).